A Characterization of Some Minihypers in a Finite Projective Geometry PG(t, 4)

Recently, Hamada and Deza [8] gave a complete characterization of all {vα + 1 + vβ + 1 + vγ + 1, vα + vβ + vγ; t, q}-minihypers for any integers α, β, γ, t and any prime power q such that q ⩾ 5 and either 0 ⩽ α = β < γ < t or 0 ⩽ α < β = γ < t where vl = (ql− 1)/(q − 1) for any integer l ⩾ 0. The purpose of this paper is to characterize all {vα + 1 + vβ + 1 + vγ + 1, vα + vβ + vγ; t, q}-minihypers for any integers t, q, α, β and γ such that q = 4 and either (a) 0 ⩽ α < β = γ < t or (b) 0 ⩽ α = β < γ < t and γ ≠ α + 1. Using those results, all (n, k, d ; 4)-codes meeting the Griesmer bound are characterized for the case k ⩾ 3 and d = 4k−1 − 4α − 4β − 4γ